Suppose, I have to prove that $A\equiv B$.
I started out by proving that $¬B \implies ¬A$. This proves $A\implies B$. Next I proved that suppose B is true and A is not and this turns out to be contradiction. So $B\implies A$. I have some feeling that I might have proved same thing thing twice. However, mathematically these seem to be sound.
Is this mathematically sound?